Permanent file Copy Sl Anthony falls. Hydraulic Laboratory

UNIVERSITY OF MINNESOTA

ST. ANTHONY FALLS HYDRAULIC LABORATORY

TechniCal Paper No. 54, Series B

On the Existence of Zero Form-Drag and Hydrodynamically Stable

Supercavitating Hydrofoils

by

R. OBA

Prepared for OFFICE OF NAVAL RESEARCH

Department of the Navy Washington, D.C.

under Contract Nonr 710(24), Task NR 062-052

November 1965

Minneapolis, Minnesotq

University of Minnesota ST. ANTHONY FALLS HYDRAULIC LABORATORY

FINAL REPORT

FLOW STUDIES ABOUT BODIES AT LOW CAVITATION NUMBERS Contract Nonr 710(24), Task NR 062-052

Prepared for

OFFICE OF NAVAL RESEARCH

Department of the Navy

Washington, D. C.

under

Contract Nonr 710(24), Task NR 062-052

MarGh 1966

Final 'Report

FLOW STUDIES ABOUT BODIEs AT LOW CAVITATION NUMBERS

Contra.ct Nc)Ur 710('21f), Task NR 062-052

Historical Summaty

This contract became effective on July 1, 1957, replacing an earlier

contract in a related area. It has been continuously active until it was

.finally terminated on July 31, 1965. During this eight-year period the con­

tract supported ex:perimental re~eal'ch conducted in the vertical free .. jet

water tunnel at the st. AnthOnY Falls Hydraulic Laboratory and some analy­

tical research as well. From August 1, 1963, to the termination, it also

supported some of the experimental research on hydrofoils conducted in the

towing tank.

The research began as general research on supercavitating flows when

the possible utility of such .flows first became apparent. However, the

emphasis has been largely on applications to hydrofoils,especially in the

later years. This emphasis resulted in a shift of support from ONR funds

to BUSHIPS funds and to the final termination of the contract. Some of the

research previously conducted under this contract is currently being supported

from Laboratory funds and the free-jet tunnel is being used on this research

as well as on other contract research.

The technical papers and pUblications prepared under Contract

Nonr 710(24) are listed in the Appendix.

Technical Summary

The first work under this contract dealt with experimentai verification

of theoretical predictions for shape of cavities and drag and lift for bodies

in a free jet. This research led to the design of a two.dimensional test

section which fitted within the originally axially-symmetric test section of

the free-jet tunnel. The results were published in Project Report No. 59

which appeared in the Journal of Fluid Me ghanics •

The early research indicated the limitations of the hybrid test

section. This was replaced by an entirely new two-dimensional test section

described in Technical Papers No. 24.B and 40-B. The two.dimensional test

-2-

section and especially its dynamometers have been modified since, but there

have been no major changes in the configuration..

The early researoh led to an attempt to study supercavitation by

injecting air into the wakes of otherwise non-c~vitating bodies. Study of

such ventilation-produced cavitation was appealing because it would permit

parallel stugies to be oonducted in the £ree~jet tunnel and in the towing

tank at the Laporatory. Both naturally occurring and ventilated cavities

could be studied i~ the new test seotion of the tunnel whereas only ventilated

oavities could be studied in the towing tank. However, three-dimensional

bodies of larger size oould be used in the tank, and also gravity acted in

the right direction. Some results of the towing tank tests (relative to tandem interference effects) are illustrated by Memorandum No. M-92 and

I .

Technical Paper No. 50.B.

One of the major discoveries of the research was the ooourrence of

pulsation in ventilated oavities onoe the cavitation number decreased beyond

a certain minimum determined by the flow conditions. The pUlsation research

is reported in Teohnical Papers No. 29.B and 32-B which have appeared in the Journal of Ship Research and in the Prooeedings of the Fourth Symposium on

Naval HydrodYnamics.

During the ventilation research it was observed that ventilation had

a material effect on noise reduotion due to cavitation. Some noise researoh

was conducted in the tunnel as reported in Teohnioal Paper No. 33-B. In

general, the tunnel has too high a background noise level to be useful for

all but the most intense noises.

The research, whioh was originally concerned with essentially mean steady flows, gradually shifted to unsteady flows associated with waves on

free surfaces, with aoceleration and deceleration of bodies, and with the

operation of flaps. The experimental resea»ch was preceded by and accompanied

by analytical research described in Technical Papers No. 34.B, 38-B, 39-B,

and 43-B in this area. Experimental results for unsteady flows associated

with a supercavitating plate oscillating at low frequencies are reported in

Technical Paper No. 49-B. Unsteady flows due to small amplitude and high

frequency oscillation of a solid flap attached to a super cavitating plate

were studied experimentally and theoretically and the results are described in Technical Paper No. 52-B.

..

One of the apparent problems when flaps are used with supercavitating

hydrofoils was the possibility that leakage through the flap hinge would have

a deleterious effect on performance. This problem was analyzed in Technical

Paper No. 51-B and, from another more general viewpoint, in an as yet

unpublished paper with the predi~ted result that leakage is probably not

an important practical problem.

Recent analytical work on the steady flow, supercavitating problem

has involved development of two-dimensional hydrofoil shapes whose lift

increases as the foil submerges further below the free surface--the so. called

stable hydrofoil. This research is described in Technical Paper No. 54-B,

but there has been no time to verify the analytical results experimentally

under the contract.

Recent experimental research has involved hydrofoils with oscillating

flaps at non-zero cavitation numbers produced largely by ventilation at

reduced frequencies up to about four. However, analysis which has proceeded

subsequent to termination of the contract is not completed and there are

some gaps in the experimental data. It is intended to submit an article

for publication in a periodical when the analysis is completed~ Some

preliminary conclusions are:

1. Flap oscillation causes cavity pressure of a ventilated cavity to oscillate along with lift, drag, and moment. Lift coefficient oscillation is larger than at zero cavitation number and first increases and then decreases as the reduced frequency is increased. This trend is opposite to the zero cavitation number case.

2. If the flap is operated at a frequency near or above the natural frequency of a ventilated cavity, the cavity may become unstable and change its regime to that of a longer or shorter cavity. There may also be a step change in the oscillating force and moment coefficients.

3. In general, cavity oscillations due to pulsation associated with ventilation and due to operation of a flap add linearity., However, at frequencies near the natural cavity frequency, they have also been observed to cancel, leaving a non-pulsating cavity_

This summary report of Contract Nonr 710(24) is being transmitted to

all those on the distribution lists for Technical Papers 52-B and 54-B.

~ . ..

APPENDIX

Reports and Publications Prep~r$d Urtder Contract 710(24)

Technical Paper No. 29-B, Instab. . VeHtilated Cavities, by Edward

Silberman and C. S. Song, 1959 also in Journal of Ship Research, Vol.

5, No.1, June 1961).

Technical Paper No. 32 ... B, Pulsation of Ventilated Cavititas, by C. S. Son~,

1960 (also in Journal of Ship Research, Vol. 5, No.4, March 1962),_

Technical Paper No. 33-B, Experimental Studies of Cavitation Noise in a. Free-Jet Tunnel, by C. S. Song, 1961.

Technical Paper No. 34-B, Unsteady. Symmetrical. Super cavitating Flows Past

a Thin Wede:e in a Jet, by C. S. Song, 1962.

Technical Paper No. 3B-B, Unsteady, Symmetrical. Super cavitating Flows Past

a Thin Wedge in a Solid Wall Channel, by C. S. Song and F. Y. Tsai,

1962. .

Pulsation of Two ... Dimensiona1 Cavities, by C. S. Song, Fourth Symposium on

Naval Hydrodynamics, Office of Naval Research, August 1962.

Technical Paper No. 39-B, A Note on the Linear Theory of Two.Dimensional

Separated Flows about Thin Bodies, by C. S. Song, 1962.

Technical Paper No. 40-B, A Dynamometer for the Two-Dimensional. Free.Jet

Water Tunnel Test Section, by Edward Silberman and R. H. Daugherty,

1962.

Technical Paper No. 43-B, A Quasi.Linearand Linear Theory for Non-Separated

and Separated Two-Dimensional. Incompressible. Irrotational Flow

about Lifting Bodies, by C. S. Song, 1963 •

.Hemorandum No. l-1 .. 92, Interference Effects for Tandem Fully Submerged Flat

Noncavitating Hydrofoils, by W. H. C. Maxwell, 1963.

Technical Paper No. 49-B, Measurements of the Unstead Force on Cavitatin

Hydrofoils in a Free Jet, by C. S. Song, 19 4.

Technical Paper Noo 50-B, Tandem Interference Effec~s for Noncavitating and

Supercavitating Hydrofoils, by J. M. Wetzel, 1965.

Technical Paper No. 5l-B, Performance of Super cavitating Hydrofoils with

Flaps. with Special Reference to Leakage and Optimization of Flap

Design, by R. Oba, 1965.

Technical Paper No. 52-B, Super cavitating Flat.Plate with an Oscillating Flap

at Zero Cavitation Number, by C. S. Song, 1965.

Technical Paper No. 54-B, On the EX~rteoce of Zero Form-Drag and Hydro­

dynamically Stable Supercavitating Hydrofoils, by R. Oba, 1965.

UNIVERSITY OF MINNESOTA

ST. ANTHONY FALLS HYDRAULIC LABORATORY

Technica;l Pa;per No. 54, Series B

On the Existence of Zero Form-Drag and Hydrodynamically Stable

Superca vi tating Hydrofoils

by

R.OBA

Prepared for OFFICE OF NAVAL RESEARCH

Department of the Navy Washington, D.C.

under Contract Nonr 710(24), Task NR 062·052

November 1965

Minneapolis, Minnesota

Reproduction in whole or in part is permitted

for any purpose of the United States Government

ABSTRACT

The linearized complex acceleration potential is obtained

for a hydrofoil of arbitrary shape in steady motion beneath a free

surface with cavity of infinite length in simple and compact form.

Under appropriate limiting conditions, it is shown that the solu­

tions obtained from this potential reduce t;:> the known solutions of'

Green for a planing foil, and of" Auslaender and Hsu for a 'flat plate

foil with or without flat flap near the free surface. (#

Using some numerical results obtained from the complex po_

tential, it is shown that there exists theoretically a supercavi­

tating hydrofoil with finite lift coefficient and zero form drag.

It is also shown that there exists theoretically a super cavitating

hydrofoil with stable characteristics when shallowly submerged; that

is, the lift coefficient increases as the submergence increase s.

Possible shapes for these hydrofoils are suggested so that the free

streamlines from the leading edges do not. intersect the foil sur­

face (the hydrofoils are physically real) and so that the pressure

on the pressure surface is everywhere greater than cavity pressure

and less than stagnation pressure (except near the leading and trail­

ing edges).

iii

CONTENTS

Abstra'ct • • • • . • • • • . , . .. .. . List of Illustrations .. , .. . List of Tables • • • • • • • List of Symbols • • • •

.. .. . . . .. . . . . . . .. . ..

I. INTRODUCTION • . . . . II. BASIC EQUATIONS .. . . . , .. • •

III. LIFI', DRAG, AND MOMENT • . . IV. RESTRICTIONS FOR THE FOIL SHAPE PARAMETERS

A. Case of Shallow Submergence B. Case of Deeper Submergence

V. EXISTENCE OF ZERO FORM-DRAG HYDROFOILS •

. .

A n

.. .. .. " . . . . ..

.. . " .. . , . .. . .. .. .. ..

. . . . . . . .. .. .. . .

VI. THE EXISTENCE OF HYDRODYNAMICALLY STABLE HYDROFOILS OPERATING NEAR A FREE SURFACE . . .. .. . ..

VII. CONCLUSIONS .. . . . . .. .. . .. . Acknowledgments • • • • • • • • • List of References Figures 1 through 16 . • • • • • • Appendix • • • • . • . . .. . .

v

. . .

Page

iii vii ix xi

1

2

8

14 15 17

17

21

24

26 27 31 45

~ OF ILLUSTRATIONS

Figure Page

1 Physical Plane, z = x + iy • • • • • • • • • • • • • • • •• 31

2

3

4

5

6

7

8

9

10

11

12

13

14

Mapping Plane by Riegels' Transformation, z = x + iy . • • •

Mapping Plane (Lower Half Plane), ~ = S :+ i11 . . , .. • • • •

Accuracy of Simple Approximation for Yf for Flat Plate;

Ao 'f 0, Al = A2 = . • • = ° · . · · · · · · · · · · · · * Change in Spray Angle ~n due to Submergence H

Various Shape Parameters A •••••••• n • •

for

• • • . . . * Zero Form-Drag Super cavitating Hydrofoil No.1 at H ~ 0;

CL ~ ¥~, CD ~ 0, (Al 'f 0, An'fl = 0) • • • • • • • • • • • ,~ '"

Zero Form-Drag Supercavitating Hydrofoil No. 2 at H = 0;

CL ~~~, CD ~ 0, (Al 'f 0, A2 'f 0, Anr.l,2 = 0) •••

Performance of AI-Hydrofoil at Deeper Submergence • • • • • •

Performance of SHl and SH2 Hydrofoils as a Function of * Submergence H • • • • • • • • • • • • • • • • •

Hydrofoil Shapes and Suction-Side Free Streamline Shapes Yf * for Various Submergences H of SHl and SH2 Hydrofoils •

Surface Velocity Distribution u * s

for Various Submergences

H of SHl and SH2 Hydrofoils . . . . . . . . . . . . , . . -

Performance of the Al Approximate Circular Arc Hydrofoil • •

Surface Velocity Distribution

Hydrofoil • • • • • • • • • •

Us of the Flat Plate

. .. .. . . . . . . . . . Performances of the A2, S-Cambered Hydrofoil • • • • • • • •

15 Relationship between Cavity Thickness T and the Change

16

* in Lift Coefficient as Related to Submergences H Flat Plate • • • • • • • • • • • • • • • • . • •

Comparison between the Calculated Spray Thickness the Experimental One by Johnson ••••••••

vii

for the . . , . . . o and . . . . . .

31

31

32

32

33

33

34

35

37

38

39

40

41

41

r'

LIST OF TABLES

Table Page ,~, .

1 Auxiliary Parameter a • . • • • • · • · • • • • · • 6

2 x as s a Function of Ss for the Foil Pressure

Side . • . • • • . • • • • • . • • • • • • • • 6

3 Auxiliary Lift Parameters M n • · • · • • • · • • • 12

4 Auxiliary Drag Parameters N • • · · n · • • · · • • 12

:5 Auxiliary Moment Parameters 0 • • · · 'n • • • • · · 13

6 Auxiliary Free Streamline Shape Parameters YfA n · • 18

\

ix

i .

e: ;::: a -ia x y

LIST OF SYMBOLS

flow aoce1eration

A, a, a1 - auxiliary parameters related only to * H

An(n;::: 0, 1, ••• )

B (n;::: 0,1, ... ) n

hydrofoil shape parameters

auxiliary hydrofoil shape parameters

CD - drag coefficient

CL - lift ooefficient

CM - moment coefficient ---. F ;::: ¢+if - complex acoe1eration potential

H - actual submergenoe

* H - modified submergenoe

I (n ;::: 0, 1, ... ) n

oavity functions

M (n ;:::

n

N (n ;:::

n

o Cn ;:::

n

YfA (n ;:::

n

0, 1,

0, 1,

0, 1,

J oorreotion factor for seoond order terms to increase the acouracyof linearized solutions

· . , ) auxiliary lift parameters

· .. ) auxiliary drag parameters

· .. ) auxiliary moment parameters

p - perturbation statio pressure due to presence of the hy­drofoil

---. q ;::: l+U-iv - normalized flow velocity taking the uniform flowveloc-

ity as unity ---. V = u-iv - perturbation velocity due to presence of the hydrofoil

0, 1, · .. ) auxiliary free streamline shape parameters

z ;::: X+iy physical plane

- x+iy z = Riegels' mapping plane

o - spray thickness

, ;::: g+ill mapping plane

p - fluid density

xi

'f - spray angle

1) = 2C - 1 - parameter ..

Subscripts

f - on the suction-side free streamline

s - on the boundary (real axis of z- and C-planes)

T - at the trailing edge

x - in the free stream direction

y - normal to the free stream direction

max - maximum

All velocities are normalized with the free stream velocity and all lengths with the chord length.

xii

ON THE EXISTENCE OF

~ FDRM-DRAG AND HYDRODYNAMICALLY STABLE

SUPERCAVITATING HYDROFDILS

I. INTRODUCTION

Increasing interest in super cavitating hydrofoils has pointed to the

need for more precise and more detailed information on the characteristics of

such foils operating near a free surface, and to the need for developing bet ..

ter foil performance.

According to the literature, Green [Ii, in a pioneer work, analyzed

performance of the super cavitating flat plate hydrofoil, a very special hydro­

foil of exceptionally high drag. Recently, Auslaender [2J analyzed the per­

formanceofthe supercavitating flapped hydrofoil composed of a flat foil and

a flat flap. Auslaender [3J and LuuandFruman [4J proposed a method to cal­

culate the hydrofoil shape for a given surface pressure distribution (inverse

method), and Johnson [5J proposed a very simple approximate method to obtain

foil performance (direct method) in which the effects of the free surface are

very roughly approximated by a single vortex in the mapping plane. In addi­

tion to these works, several experimental studies have also been published.

At this stage, however, the performance of presently available super­

cavitating hydrofoils operating near a free surface is not necessarily good

enough to apply to a practical high-speed surface craft and there remain some

basic difficulties. The first difficulty concerns the hydrodynamic stability

of the hydrofoil. Most of the known supercavitating hydrofoils are unstable

when operated under a free surface; that is, the lift decreases as the sub­

mergence becomes greater. Consequently, a supercavi tating hydrofoil boat may

be unstable in rolling, pitching, and heaving motions. Two well-known meth­

ods of providing stability are (1) touse dihedral foils, and (2) to use con­

trol devices. The first method will result inevitably in higher drag and

lower efficiency. The second method introduces an additional technical prob­

lem, the unsteady characteristics of the control devices themselves.

The second basic difficulty is that the efficiency (lift-drag ratio)

of super cavitating hydrofoils is, in general, not as high as that of fully

INumbers in brackets refer to the List of References on page 27.

2

wetted hydrofoils and airfoils. A third difficulty arises in that the un­

stead-y- characte-I'-iBtics a-Llow dra~-cambered-h;¥'dX'Ofoil~nnot be __ pl'Bdicted.

In this report, possibilities for removing the first and second dif­

ficulties given will be discussed theoretically. First, by improving John­

son's approximate method [5J, an accurate method to estimate the performance

of super cavitating hydrofoils of quite arbitrary shapel operating at quite

arbitrary submergence is proposed. After rather complicated numerical compu­

tations it will be shown that a hydrodynamically stable super cavitating hydro­

foil exists for which lift increases with submergence for small submergence.

The theory also indicates the existence of an infinite number of zero form­

drag hydrofoils with finite lift satisfying the necessary and sufficient con­

ditions for the supercavitating flow of positive cavity thickness and posi­

tive surface pressure on the pressure side.

II. BASIC EQUATIONS

The two-dimensional, incompressible, inviscid flow around a supercav-

itating hydrofoil of arbitrary shape y (x). operating near a free surface, s

in the physical plane z = x + iy, is shown in Fig. 1. To simplify the pres­

ent problem, it is assumed that the foil chord is unity, the leading edge is

located on the origin of the coordinate system, and the uniform flow velocity,

taken as unity, is the normalizing velocity for all velocities. These assump­

tions do not limit the scope of the pr~sent problem. In this paper discussion

will be limited to the special infinite trailing cavity case; also infinite

Froude number is assumed.

The flow velocity q at any point may be expressed as the sum of the

uniform velocity of unit magnitude and the perturbation velocity due to pres­

ence of the hydrofoil u(x, y) - iv(x, y), as follows:

-4 -4

q = 1 + u(x, y) - iv(x, y) = 1 + V

Assuming that u, v «1 and neglecting second order terms, the Euler­

ian equation of motion may be expressed as

lAs shown later, the hydrodynamically stable hydrofoils have larger cam­ber near their trailing edges. For such hydrofoils, it is doubtful that John­son's approximate solution [5J would apply.

o 0)-4 1 (ot + OX q ~ - p grad p ~ grad ¢

where ¢(x, y, t) ~ - £, p is the constant fluid density, p

turbation pressure due to the presence of the hydrofoil, and ¢ acceleration potential.

3

(2)

P is the per­

is the Prandtl

The equation of continuity and the assumption of u, v« 1 lead to

the following result for the acceleration a ~ a - ia .: x y

div a: ~ 0

Then the acceleration potential ¢ satisfies the Laplace equation,

(4)

Therefore, a conjugate function ~ and f1 complex acceleration pote:ntial --+ F(x, y, t) may be defined as follows:

-4

F(x, y, t) = ¢ + i~ (6)

Then

--+ dF -4

dz = a = ax

For the steady case

--+ -4

dF dV -=-dz dz' ¢ + i~ = u - iv (8)

Next, boundary conditions are considered. By Riegels' transformation

[6J the present rather complicated boundary value problem shown in Fig. 1 can

be reduced to a rather simple boundary value problem on the z-plane, shown in

Fig. 2, in which the boundary values are given only on straight slits CD, OD,

and OB. Here, the following relation holds between the boundary values in

the z-plane and those in the z-plane:

4

-4

1 + F (z) -4 s

1 + F (z) = -----:::-­S

s

where the subscript s means "on the boundary" and

slope of the boundary line. If us' v s ' (~) « 1, s

is the

(10)

The modified submergence

has still not been determined.

* H

* H

in the mapping plane z, shown in Fig. 2,

is not necessarily equal to the actual

submergence, H, since H does not indicate the mean free surface level, ex­

pecially for the flat plate hydrofoill at small submergence with a rather

large spray angle (see Fig. 5). However, since low drag, cambered hydrofoils

* generally have rather small spray angles, the assumption H ~ H might be

roughly applied to such cambered hydrofoils.

Finally, the boundary conditions under the assumption of small pertur­

bation may be summarized as follows:

1. Assuming that the disturbance pressure p due to presence of the hydrofoil is zero on the free sur­faces, it follows that

2.

- ?O - + x - , Ys = 0 s

¢s (x) ¢ (x) 0 for - ?l - 0-= = x - , Ys = s s H*-- --00 < x < 00, y = s s

On the foil surface ary condition is

y s(x), the linearized bound-

v = _=-s _ = _ if; ex) + 0(E 2)

1 + u s s

3. The condition at upstream infinity z = z = -00 is

(11)

(12)

1 *-In the Appendix the useful result H = (), where () is the spray thick-ness (see Fig. 1), was found even for the extreme flat plate case. Also in the Appendix the relation between () and the submergence H is given.

5

~

F( -a:> ) = 0

4. The Kutta Joukowsky condition is to be enfo~ced.

Then F('z) is continuous at z = zT'

These basic equations o~ bounda~y conditions may be ~eadily gene~al­

ized to the unsteady p~oblem fo~ hyd~ofoils ope~ating nea~ the f~ee su~face,

that is, the third p~oblem mentioned in the Int~oduction. This p~oblem will

be discussed in a succeeding repo~t. The above bounda~y conditions a~e quite * simila~ to the ones given by Johnson [5J if the assumption H = H can be

accepted.

Now that the boundary conditions have been applied to the mapping plane

Z, the p~oblem solving can begin. Fi~st, the z-plane is t~ansfo~med into the

lower half C-plane by the mapping function [5J,

z = A[C - a log(l + ')J a (14)

He~e auxiliary pa~amete~s A

to the modified submergence

and a have been introduced; these a~e ~elated * H as follows:

* H = TTaA, t = 1 - a ~n(l + :)

The problem has now been ~educed to a ve~y simple bounda~y value 'p~oblem in

which all boundary values a~e given only on the ~eal S-axis.

Equations (9) ahd (14) result in

_ Ss x = x = A[s - a ~n(l + --)J s s s a (14' )

* Some numerical data fo~ a vs H and x vs S a~e p~esented in Tables 1 s s and 2. More such data can be found in Ref. [7J.

~

Since the complex acceleration potential F(z) must satisfy the lin-

ea~ Laplace equation, the gene~al solution may be easily obtained by supe~­

posing partial solutions and may be exp~essed as follows [8J[9J:

~ ~ -+ F(z) = F(z) = F(C)

a:>

= i :E A I n=O n n

(16)

6

TABLE 1

Auxiliary Parameter a

* * H a H a

0 0 4 0 • .5484.5 0.2.5 0.06.5098704 .5 0.63.547 0 • .50 0.11711674 6 0.71.510 0.7.5 0.16242041 8 0.8.5821 1 0.2032.555 10 0.98565 2 0.339830 100 3.67608 3 0.4.51460 00 00

TABLE 2

x as a Function of Ss for the Foil Pressure Side s

Ss

* * * * * * ,~

x H =0 H =0.2.5 H =0 • .5 H =1 H =2 H =.5 H =00 s

0.00001 0.00001 0.00102 0.00131 0.00161 0.00190 0.0022.5 0.00316 0.0001 0.0001 0.0033 0.0042 0.00.51 0.0060 0.0072 0.0100 0.001 0.0010 0.0108 0.0136 0.0165 0.0194 0.0228 0.0316 0.01 0.0100 0.0383 0.0466 0.05.53 0.0638 0.0739 0.1000 0.02 0.0200 0.0.576 0.0689 0.0808 0.0924 0.1061 0.1414 0.0.5 0.0500 0.1024 0.1188 0.1361 0.1530 0.1729 0.2236 0.10 0.1000 0.1636 0.1843 0.2062 0.2276 0.2.526 0.3162 0.20 0.2000 0.2703 0.2943 0.3199 0.3448 0.3739 0.4472 0.30 0.3000 0.3689 0.3931 0.4190 0.4443 0.4738 0.5477 0.40 0.4000 0.4636 0.4864 0.5108 0.5348 0 • .5627 0.6325 0 • .50 0 • .5000 0 • .5.558 0.5762 0.5981 0.6196 0.6447 0.7071 0.60 0.6000 0.6465 0.6637 0.6823 0.7005 0.7218 0.7746 0.70 0.7000 0.7360 0.7495 0.7641 0.7784 0.7952 0.8367 0.80 0.8000 0.8247 0.8340 0.8441 0.8540 0.8657 0.8944 0.90 0.9000 0.9126 0.9174 0.9226 0.9278 0.9338 0.9487 0.95 0.9500 0.9.564 0.9588 0.9615 0.9641 0.9671 0.9747 1.00 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

where the A are defined in Eq. (19) below and n

I ~ &r ~ u--:--o = 1 ~ ~ v+l ' 11 = v ~ ~v ~ 1 ,

v = 2, .. 1

7

(18)

and where I and I are the Hilbert transform of cot -26 and sin n9 shown on>

in Eq. (16") below. On the free surface or on the cavity boundary I, v 1= 1

and In is real. Then

(16' )

On the foil surface, ~l ~ v ~ 1. Then

_ _ _ 6 CD CD

F (x ) = F (x ) = F (g ) = A cot - +:EA sin n6 .. i(-A +:EA cos n6) s s s s s s 0 2 1 n 0 1 n

~ = 12 (1 - cos e), '='s

~ -> > o = Q = -TT

F (-CD ) = 0, and s F(c) is continuous at v = 1 (at the trailing edge where

z = 1). Therefore, s -the solution F(C) satisfies boundary conditions 1, 3, and 4 list~d above.

On the physical plane z the hydrofoil

slope (gxd) and then ~ (x) and ~ (g ) x s s s s s

shape y (x) s

are known.

is given. The foil

Thus the arbitrary

constants Ao' Al ,

and the submergence,

A2 , • • ., which are connected only with the foil shape

are determined easily as follows:

t/; (g) de s

-TT

A = g f t/; (g) co s ne de, n = 1, 2, 3, . • . n TT s o

8

A flat plate foil is given by Ao f. 0 and An-fO = O. The greater the number

n of A, the more cambered is the foil near the leading and/or the trailing n edge.

III. LIFT, DRAG, AND MOMENT

The lift, drag, and moment per unit span of foil, respectively, may be

expressed in coefficient form as follows:

where J is a

cipally to the

1

CL = - ~ f ¢ s (x) dx s o

1

CM = - J22f xs ¢s(x) o

dx s

dx s

(20)

(22)

correction factor to offset the second order errors due prin­

assumption that the Riegels' factor 1 in Eq. (9)

J 1 + (9:£.)2

" dx s

is unity. From the studies of the arbitrary hydrofoil in infinite fluid [lOJ

and of the flat plate hydrofoil at arbitrary submergence (see the Appendix),

the values of J may be expressed approximately as follows:

The lift coefficient CL is

C 1 + -1

J = _~2;;:.. + o(i) cos A o

where

and

.. TT

A f (A 9 ~ A ) (1 - cose)sin9 dO C = - - cot-2 + £..J. sin n6 0 L J 0 1 n 1 + 2a - cose o

Q)

;;lL:M A Jon n

> n = 2

a1 = 1 + 2a .. 2a ~ 1 + :

The drag coefficient CD is

-TT

C =!:. f (A cot-2e + f A sin n9)(-A + fA cos n9)(1 .. cos9)sin9 de D J 0 1 n 0 1 n 1 + 2a - co s9

o

where

9

(20' )

(24)

(21' )

10

No = TIAa1 ,

The moment coefficient 1M is

> n = 2

BS = -AoAS + A1A4 + A2A3 1 2

B6 = -AoA6 + AlAS + A2A4 + 2A3

B7 = -AoA7 + A1A6 + A2AS + A3A4

-TI C - _ A2af{1-COSQ _ .R,n(l-cosQ + I)} (A cot~ + fA sin ne) (l-cose)sine de M - J2 2a 2a 0 2 1 n 1 + 2a - cose

o

n ill

+2: n+1

n a1 n+1

n-1 n a1

n-1

CD a1 A-SaL -A n n n n=l

2A+n ill n n a1 CD

- 4a 2: 2: A(A+n) A - 2a 2: n=l A=l n n=3

1 ill ;; -2: 0 A J2 0 n n

where

n/=2A

n=2,4, •••

(22' )

n+l n-l _ TTaA21 *. n na.l nal

On - 2 -T al + --"'n~+l- + --"'n=--l-

° = TTaA -T*a n + nal + nal 21 n+l n-l

n a 1 n+l n-l

* T = 2 + 4a ~n4aal

n 2A+n 8aal n 4anal

n - t:l A(A+n)

for even n ~ 4 2A+n

n 4anal

- El x(A+n)

for odd n ~ 3

n-l 2 n f ana El A(n-A)

11

The coefficients Mn' Nn' and On introduced in Eqs. (20'), (21'), and

(22') are independent of the hydrofoil shape; these have been calculated and

are shown in Tables 3, 4, and 5. These tables show reasonably good conver­

gence in the series for M, N, and 0. The series A is generally a n n n n good convergent series because the hydrofoil form is usually very smooth.

Therefore, though the numerical estimations in Eqs. (20'), (21'), and (22')

seem at first to be very complicated ones, they may be evaluated readily in

the practical case.

* For a special limiting case of H ~ CD, that is, H~ CD,

(26)

(28)

TABLE 3 ~ !'0

Auxiliary Lift Parameters M n

* M ~ M2 M3 M4 M5 M6 M7 M8 M9 H 0

0 3.1416 1.5708 0 0 0 0 0 0 0 0 0.25 2.3179 1.6184 -0.1821 -0.1099 -0.0664 -0.0400 -0.0242 -0.0146 -0.0088 -0.0053 0.50 2.1808 1.6238 -0.2609 -0.1333 -0.0681 -0.0348 -0.0178 -0.0091 -0.0046 -0.0024 0.75 2.1048 1.6251 -0·3117 -0.1421 -0.064(; -0.0295 -0.0135 -0.0061 -0.0028 -0.0013 1 2.0537 1.6251 -0.3485 -0.1455 -0.0607 -0.0254 -0.0106 -0.0044 -0.0018 -0.0008 2 1.9429 1.6222 -0.4359 -0.1439 -0.0475 -0.0157 -0.0052 -0.0017 -0.0005 -0.0002 5 1.8256 1.6138 -0.5384 -0.1249 -0.0290 -0.0067 -0.0016 -0.0004 -0.0001 0 10 1.7585 1.6062 -0.6009 -0.1042 -0.0181 -0.0031 -0.0005 -0.0001 0 0 100 1.6343 1.5858 -0.7218 -0.0432 -0.0026 -0.0007 0 0 0 0 Q) 1.5708 1.5708 -0.7854 0 0 0 0 0 0 0

TABLE 4

Auxiliary Drag Parameters N n

* N Nl N2 N3 N4 N5 N6 N7 NS N9 H 0

0 3.1416 -1.5708 0 0 0 0 0 0 0 0 0.25 2.3179 -0.6995 -0.7368 -0.4447 -0.2684 -0.1620 -0.0978 -0.0590 -0.0356 -0.0215 0.50 2.1808 -0.55'70 -0.8059 -0.4117 -0.2103 -0.1074 -0.0549 -0.0280 -0.0143 -0.0073 0.75 2.1048 -0.4797 -0.8338 -0·3800 -0.1732 -0.0790 -0.0360 -0.0164 -0.0075 -0.0034 1 2.0537 -0.4287 -0.8479 -0.3540 -0.1478 -0.0617 -0.0258 -0.0108 -0.0045 -0.0019 2 1.9429 -0.3207 -0.8656 -0.2857 -0.0943 -0.0311 -0.0103 -0.0034 -0.0011 -0.0004 5 1.8256 _0.2118 -0.8637 -0.2004 -0.0465 -0.0108 -0.0025 -0.0006 -0.0001 0 10 1. 7585 -0.1524 -0.8529 -0.1478 -0.0256 -0.0044 -0.0008 -0.0001 0 0 100 1.6343 -0.0491 -0.8142 -0.0489 -0.0029 -0.0002 0 0 0 0 Q) 1.5708 0 -0.7854 0 0 0 0 0 0 0

13

TABLE 5

Auxiliary Moment Parameters ° n

H* ° °1 °2 03 °4 °5 °6 0

0 0.7854 0.7854 .. 0·3927 0 0 0 0 0.25 0.6864 0.7743 .. 0.4638 0.0318 0.0123 0.0066 0.0022 0.50 0.7002 0.7675 .. 0.5163 0.0536 0.0166 0.0053 0.0016 0·75 0.7014 0.7625 .. 0.5375 0.0698 0.0238 0.0047 0.0010 1 0.7061 0.7586 .. 0.5531 0.0775 0.0189 0.0040 0.0009 2 0.7351 0.7482 .. 0.5876 0.1088 0.0173 0.0015 .. 0.0002 5 0.8253 0.7342 .. 0.6265 0.1614 0.0093 .. 0.0024 .. 0.0011 10 0.9477 0·7231 -0.6485 0.1918 .. 0.0013 -0.0034 .. 0.0011 100 0.5074 0.7014 -0.6667 0.2684 -0.0260 -0.0043 -0.0002 ro 0.4909 0.6872 .. 0.6872 0.2945 -0.0491 0 0

These solutions agree exactly with the accurate modified linearized solutions

for an arbitrary form hydrofoil given in the author's previous work [10J. If

the correction factor J can be assumed to be unity, these solutions agree

with the Tulin .. Burkhart solutions [llJ.

* For a special limiting case of H ~ 0,

C = II (A + ~) L J 0 2

(30)

If J = 1, the solutions agree with those for a planing craft [12J. It may

be observed that, in this case, the drag coefficient CD is connected only

with the shape parameter Ao; (the drag corresponds to the so-called spray

drag [12J).

For a special case of the flat plate hydrofoil,

14

TI 2 C =-Aa A D J 1 0

(34)

These seem to agree with Green's exact solutions [lJ up to second order terms,

when compared on the basis of spray sheet thickness (see the Appendix). If

J = 1, they agree with the solutions of Auslaender [3J and Hsu [13J.

IV. RESTRICTIONS FDR TEE FDIL SHAPE PARA.ME'l'ERS A n

In this section the restrictions upon the foil shapes or the incidence

angles, that is, the parameters A and A, are discussed. A physically o n

meaningful solution should satisfy at least the following two conditions.

1.

2.

The absolute value of the surface velocity on the pressure side I qsl not only needs to be less

than that of the cavity surface velocity (the as­sumed magnitude of which is unity) to avoid cavi­tation on the pressure side, but also needs to be more than zero. As the lift force on a super­cavitating foil is induced only by the positive pressure p on the pressure side of the foil, s it follows from the condition I q I ~ 0 ( that is, s

~l Ps max - 2'P coefficient

q 2) that there exists a maximum lift (])

CL max for each hydrofoil. In the 1 2

limiting case of qs = 0, Ps = 2'P q(]) over the

entire chord, C = 1. L max

The distance between the vacuum-side free stream­line and the foil surface line, that is, the so-

called cavity thickness, should be positivel at any point along the chord.

lFor example, the cavity thickness of Tulin' s profile (Ao = 0) [llJ,

which partly satisfies condition 1, is largely negative over the chord. Also the actual performance is quite different from that predicted because of an increased incidence angle to insure positive cavity thickness over the chord. Consequently, the predicted performance of Tulin's profile (A = 0) seems to be quite meaningless physically. 0

15

The author believes that any discussions or any theoretical results ignoring

these two important physical conditions for a given arbitrary shape hydrofoil

operating at an arbitrary submergence are quite meaningless until these con­

ditions have been checked.

Condition 1 can be checked easily. For the shape parameters A and o

A the surface velocity distribution expressed by,Eq. (16") can be obtained n by simple harmonic analysis (cosine series). Although condition 1 has been

checked by several investigators [5J[9J[11J in the course of their work, there

still remain some interesting problems concerning it. For example, this con­

dition refutes the possibility of Weiker's idea [14J that CL be finite while

CD = 0 for the S-camber hydrofoil (A2 f 0, Ao = A3 = A4 = ••• = 0 at H':< = co ).

To check condition 2 the suction-side free-streamline shape Yf must

be calculated through a rather complicated pro,cedure; consequently most pre­

vious studies have either ignored it or performed a very superficial calcula­

tion. The following paragraphs describe the calculation of the free-streamline

shape Yf'

A. Case 2f Shallow Su.bmergence

Generally speaking, the shallower the submergence >:-:

(H < 1),

fects are

the more influential the submergence ef­

on the foil performance [lJ[2J. Thus, it

is necessary to calculate many points in

order to define the relationship.

* H < 1 in

The free-streamline shape Yf may be expressed

as follows:

x s

Yf = f vf ds o

CD J-Ss I S A LA ~~ ds o n a ':>

o

To simplify the relation, for shallower submergence, * . consider that H < 1, a« 1, Is I <..< 1, and let

s I (I; ) ~ I (-a). Therefore, Yf may be expressed n s n approximately by the following straight lines:

16

where

2t I n

y == lfA I Ca) I x f 0 n n

- 1,

I l' n- t == 2a + 1

co Practically speaking, the factor I: A I is the ver-o n n tical velocity component on the spray jet at infinity,

or approximately the spray anglel 'f. That Eq. (35')

is sufficiently accurate for the flat plate (where

Ao =/: 0, An=/:O == 0) is demonstrated * in Fig. 4 where

it may be seen that the smaller H is, the better

the accuracy is. The accuracy for the parameters

A (n =/: 0) seems to be better, for the shapes of the n

free streamlines y f are closer to being straight

lines when calculated byEq. (5) as shown in Fig. 10.

The nature of I in Eq. (:35') is shown in Fig. 5 for n tan 'f

several typical shape parameters An (In == -"'A-n~ for n

each A). It is apparent from this figure that there n

is an exceptionally large spray angle 'f for the o

fl ~~t IPlate (n == 0). Also there is a tendency for

jH~ to approach zero rapidly with an increase in

* the submergence H and the subscript n of An.

Therefore, hydrofoils having higher n-terms of A n

(that is, hydrofoils greatly cambered at the leading

and/ or the trailing edge) might be expected to have

* flat CL vs H performance (except for the extreme * ...... case of H == 0).

lThe tangential component l+u s is close to unity.

B. Case of Deeper Submergence

The accuracyofEq. (3.5') deteriorates with an in­

* crease in the submergence H; consequently Eq. (3.5') :-:<

would be of no use for H > 1. However, for such

cases, Eq. (3.5) is still applicable. After several

trial calculations the following fairly accurate rep­

resentation for Yf was found:

The values YfA' YfA' and YfA are tabulated in o 1 2

Table 6. Furthermore, the form Yf for xs > 0 • .5, and Y fA (n > 2) are very clo se to straight lines

parallelnto the ones definedbyEq. (3.5'). Using Eq.

(3.5"), calculations to check condition 2 may be per­

formed very easily.

V. EXISTENCE OF ZERO FDRM.DRAG HYDROFOiLS

17

(3.5")

Many investigators have explored the possibility of using a supercav­

itating hydrofoil in high speed machinery. At present, however, the appli­

cation to such machinery is extremely limited. The main reason for this is

believed to be lower efficiency associated with high drag-lift ratio. Un­

economical performance has been considered inevitable for hydrofoils with

rather large trailing cavities and rather large wakes. In this section, how­

ever, it is shown that uneconomical performance may be eliminated for hydro­

foils operating near a free surface; however, the submergence effects on hy­

drofoil performances may be extremely large.

>:< The lift and drag coefficients CL and

H -+ 0, are shown in Eqs. (29) and (JO). In

CD' at small submergences of

these equations, if A = 0 o

without restriction on the other A, n

obtained with the lift coefficient

quite general hydrofoil shapes may be ........ n

CL = '2 Al , and the drag coefficient

18

x s

o 0.01 0.02 0.03 0.05 0.07 0.10 0.20 0.30 0.40 0.50 0.60 0.'70 0.80 0.90 0.95 1.00

x s

o 0.01 0.02 0.03 0.05 0.07 0.10 0.20 0.30 0.40 0.50 0.60 0.70 O.RO 0.90 0.95 1.00

TABLE 6

Auxiliary Free Streamline Shape Parameters YfA n

* * H = 1.0 H = 2.0

o 0.0515 0.0851 0.1141 0.1655 0.2118 0.2754 0.4625 0.6311 0.7906 0.9448 1. 0957 1.2443 1.3914 1.5374 1.6101 1.6827

o 0.0416 0.06'18 0.0902 0.1289 0.1630 0.2089 0.3382 0.4485 0.5486 0.6418 0.7300 0.8146 0.8961 0.9753 1.0141 1.0524

o 0.0071 0.0133 0.0193 0.0306 0.0412 0.0567 0.1050 0.1539 0.1951 0.2387 0.2816 0.3243 0.3667 0.4089 0.4299 0.4510

o 0.0067 0.0123 0.0176 0.0273 0.0365 0.0496 0.0887 0.1242 0.15?? 0.1895 0.2200 0.2497 0.2787 0.3070 n.3209 0.3347

o 0.0050 0.0089 0.0124 0.0189 0.0245 0.0325 0.0558 0.0831 0.0965 0.1155 0.1339 0.15?1 0.1701 0.1879 0.1968 0.2057

o 0.0046 0.0077 0.0105 0.0151 0.0193 0.0251 0.0402 0.0528 0.0643 0.0745 0.0834 0.0923 0.1007 0.1087 0.1125 0.1162

x s

o 0.01 0.02 0.03 0.05 0.07 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 1.00

x s

o 0.0045 0.0137 0.0319 0.0625 0.1082 0.1707 0.2500 0.3444 0.4503 0.5625 0.6747 0.7797 0.8705 0.9406 0.9849 1.0000

o 0.0463 0.0760 0.1015 0.1461 0.1857 0.2397 0.3949 0.5309 0.6567 0.7761 0.8911 1.0028 1.1121 1.2194 1.2725 1.3252

H*

o 0.0191 0.0415 0.0740 0.1162 0.1669 0.2245 0.2868 0.3514 0.4160 0.4779 0.5348 0.5846 0.6254 0.6557 0.6744 0.6807

o 0.0068 0.0128 0.0185 0.0289 0.0389 0.0531 0.0968 0.1373 0.1758 0.2133 0.2498 0.2831 0.3145 0.3559 0.3733 0.3904

= <D

o 0.0030 0.0080 0.0166 0.0291 0.0454 0.0651 0.0874 0.1115 0.1362 0.1605 0.1832 0.2033 9.2200 0.2324 0.2402 0.2427

o 0.0046 0.0083 0.0115 0.0169 0.0220 0.0287 0.0478 0.0642 0.0790 0.0931 0.1064 0.1142 0.1189 0.1440 0.1500 0.1557

o 0.0020 0.0047 0.0088 0.0139 0.0197 0.0259 0.0322 0.0384 0.0443 0.0494 0.0540 0.0578 0.0609 0.0632 0.0645 0.0649

i '-'-

19

CD ;- 0, if the surface pressure on the pressure side and the cavity thick­

nessl may be made positive over the chord. Also, there is a maximum value of

C1 determined by condition 1 on page 14 (see Ref. [9]) which cannot be ex­

ceeded; this will be called C1 . max

Figure 6 shows the profile y s' the free streamline y f' and the sur­

face velocity distribution us/AI of the so-called Al-hydr~foil (~ i= 0,

A :::: A :::: A :::: , •• - 0) operating in the neighborhood of H"':::: 0. As seen 02:3

in Fig. 6, the cavity thickness and the

the chord. If C1 is restricted to C1 < surface pressure are positive over

C1 (or Al < Al ), the nec-max max essary and sufficient conditions 1 and 2 for supercavitating flow are satis-

fied completely. Here, theoretically, one could find a zero form-drag hydro­

foil of C1 :::: ¥ Al (finite) and CD:::: 0. For the hydrofoi~"defined by A2/Al ::::

0.5, Al f. 0, A2 f. 0, Ao:::: A:3 :::: A4 :::: ••• :::: ° at H'" :::: 0, the cavity n

thickness and the surface pressure are also positive over the chord, C1 :::: '2 Al

(finite), and CD ';;; ° (see Fig. 7). Thus this hydrofoil is also theoretically

a zero form-drag hydrofoil and has a larger included angle at the leading edge

than the one for the AI-hydrofoil. In the same way, an infini te number of

zero form-drag hydrofoils might be found. Auslaender DJ also shows the the­

oretical po ssibili ty of the existence of hydrofoils such as CD ~ ° and

CL > ° (finite) at H':<:::: 0. However, he did not fully consider the necessary

and sufficient condition 2 for the supe~cavitating flow.

Now, the discussion will be extended to the case of arbitrary submer­

gence. To arrive at the approximate tendency in the C1 vs H * performance

or the CD vs l performance, the discussion will first be limited to a spe­

cial case of Al f. 0, A :::: A :::: A :::: ••• :::: 0. For this case, 02:3

CD ---!!A 2 -2 1

As seen in Fig. 8 for this AI-hydrofoil family, the lift scarcely changes with >:<

depth staying within 3.5 per cent for the whole range of H, but the drag

lCavity thickness is defined as the difference between the foil surface on the pressure side and the vacuum-side free streamline.

20

* changes greatly. For given lift coefficient, the drag coefficient at H = (J)

>:' reduces to about 1/2, 1/5 ,and 0 for H = 1, 0.25, and 0 respectively as

may be seen in Fig. 8. For the AI-hydrofoil, the cavity thickness and the >:.::

surface pressure are positive over the chord, unlE)ss H is too large; (if

* H = ro, the cavity thickness becomes slightly negative near the leading

edge, as shown in Ref.· [9J). Thus the necessary and sufficient conditions 1

and 2 for the supercavitating flow seem to be satisfied unless ~ is too

large. Figure 8 indicates that:

1. The submergence effects on the performance of hy­drofoils operating near a free surface are ex­tremely large and cannot be neglected. CL and

>:<

2.

CD at H = 1 '" 10 of the practical usable ra~g:

differ only moderately from CL and CD at H ro.

The drag coefficient CD for the Al-hydrofo~l fam­

ily rapidly decreases as the submergence H ~ O. This peculiar tendency is completely opposite the tendency of the flat plate. CL scarcely changes in this range also.

The AI-hydrofoil is a special shape and the above discussion is not

necessarily relevant to a general discussion of submergence effects on hydro­

foil performance. To increase the generality, define two typical cambered

hydrofoils, for example, the 3Hl~ndthe 3H2 hydrofoilsl , made up of circular

arc shapes (AI f 0, A = A - A - - 0 at H':' = 0), 3-shapes (A f 0 o 2- J-',~"- 2 ' Ao = Al = AJ = A4 = .•• = 0 at H = 0), and flat inclined shapes (Ao f 0,

Al = A2 = • • . = 0) for the purpose of satisfying the necessary and suffi­

cient condition 2 for the super cavitating flow. The ratio A2 /Al for 3Hl or

3H2 is taken as -1/2 or 1/2 respectively, while A = O. Performance is shown o

in Fig. 9. The hydrofoil shapes y s and the suction-side free streamline

shapes Yf are shown in Fig. 10, while the surface velocity distributions

u are shown in Fig. 11. These figures indicate that: s

1

1.

formances vary to a considerable degree with the hydrofoil shape.

2. Three quite different types of behavior are pres­ent in the CD vs H* performance: one in which

The performance of the 3H2 hydrofoil was also discussed in the previous section (see Fig. 7).

i I

4.

* cD decreases with an increase in H, as in the

flat plate; '" the second in which CD is quite le­

vel over H'" except in the neighborhood of H*::= 0, as in the SIn hydrofoil; and the third in which CD increases with H*, as in the SH2 Hy-

drofoil.

The change in CL due to change in submergence

is attributable principally to the change in the surface pressure on the forepart of the foil, x < 0.5 (see Fig. 11).

s * The CL vs H performance of the SHl hydrofoil

is especially interesting since CL is approx-~:<

imately constant for H > 2.

5. The results for CD vs H* and CD/CL obtained

for the AI-hydrofoil family are apparently veri­

fied for these somewhat more complicated shapes.

21

For both SHI and SH2 hydrofoils the necessary and sufficient condi­

tions 1 and 2 for supercavitating flow are completely satisfied except for

* * the extreme submergence raUf6e of H"> 5 (for SH1) and H ;' (]) (for SH2) re-

spectively, if CL < CL • max

Johnson [5J discusses in his approximate theory the camber effects

upon the CL vs H':< performance. However, he did not fully consider the phys­

ically necessary and sufficient conditions for superdavitating flow. Gener­

ally speaking, every supercavitating hydrofoil has its original minimum in­

cidence angle [9J, a . (and A .), to satisfy condition 2. As seen in mln 0 mln Fig. 9, the submergence effects for the lift coefficient on a flat plate are

quite large. Therefore, it is apparent that Johnson's results should be mod­

ified to include the effect of submergence on CLA and to satisfy condition

2. o

VI. THE EXISTENCE OF HYDRODYNAMICALLY STABLE HYDROFOILS OPERATING NEAR A FREE SURFACE

Most of the known two-dimensional supercavitating hydrofoils (for ex­

ample, the flat plates, the sm hydrofoil, the SH2 hydrofoil, the circular

arc hydrofoil, the Tulin hydrofoils, and the flapped hydrofoil) are hydrody­

namically unstable when operated under a free surface, especially at shallow

submergence where low form drag is to be expected; that is, the lift decreases

22

as the submergence becomes greater. Consequently, a supercavitating hydro­

foil boat may be unstable in rolling, pitching, and heaving motions.

Of course, several practical methods to provide stability in such hy­

drofoil boats have been proposed, as for example dihedral foils or control

devices. However, no research seeking an inherently hydrodynamically stable

hydrofoil form has been reported. In this section, therefore, as the first

step in such direction, the theoretical possibility of the existence of such

stable two-dimensional super cavitating hydrofoil forms is discussed.

The following discussion is limited to the shallower * H range in which

CL changes considerably, as shown, for example, in Fig. 9a. First, consid­

ering a hydrofoil of appro::imate circular arc camber (AI, i= 0, Ao = 0.1 Al ,

A2 = A3 = • . • = 0 at H'" = 0) called Al , the CL vs H'" performance was

calculated and is shown in Fig. 12. This figure also shows the hydrofoil shape

y s' the suction-side free streamline shape y f' and the surface velocity

distribution u. The restriction A = 0.1 Al resulted from the necessary s 0

and sUffi"cient condition 2 for the supercavita ting flow. As can be seen in

Fig. 12, if the restriction CL < CL max is used, the Al -hydrofoil completely

satisfies the necessary and sufficient conditions 1 and 2, in the working >,~

range of H < 1.

In Fig. 12, the CL vs H performance, although it still has unstable

characteristics, is much better than the performance of the flat plate. The

* improvement in the CL vs H performances probably correspond to the large

difference in the surface velocity distributions which may be seen by compar­

ing Fig. 12b with Fig. 13 for the flat plate. Since the change in CL due

* to H princ.ipally corresponds to the change in the surface velocity in the

fore part of the foil, as shown in Figs. 12b and 13, itis quite natural that

the change in CL is exceptionally large for the flat plate in which velocity

distribution is one-sided toward the front, and fairly small for the Al-hydro­

foil (in which velocity distribution is approximately leV\el). Furthermore,

the large difference in the CL vs H'~ performances may correspond to the

large differences in the spray angle T shown in Fig. 5; (the change in CL

corresponds to the change in momentum difference between the upstream infin->:<

ity and downstream infinity, that is, to the product H x T).

Returning to the CL vs H'~ performance of the Al-hydrofoil, if the

narrower working range of H~' < 0.25 is selected, the most severe restriction

23

on the cavity thickness (condition 2) may be greatly reduced, and consequently

the CL vs H* performance maybe improved as shown in Fig. 12c. At the lim-

* * iting condition H ~ 0, the CL vs H curve tends to a maximum value of

CL, Hence, this Al-hydrofoil does not have a stable characteristic, even in

the limiting case.

The performance calculated for an A2-hydrofoil of a typical S-camber

* (A2 < 0 , A ~ A ~ A 013

~ • • . ~ 0 at H ~ 0) is shown in Fig. 14. With a

locally negative cavity thickness and negative static pressure as shown in

Fig. 14, the A2-hydrofoil itself is physically meaningless, although it pos­

sesses a stable lift characteristic. However, a physically meaningful hydro­

foil can be obtained by combining the A2-hydrofoil with a known hydrofoil. Al

In an attempt to combine the A2-hydrofoil and Al-hydrofoil, A ':' ~ 1 2 H ~O

is assumed for the camber to simplify the problem. In the combined hydrofoil

shown in Figs. 12 and 14 the necessary and sufficient coDditions 1 and 2 for >:<

super cavitating flow are completely satisfied in the working range of 0 < H

< H ~", where H >:'< 0.25. It should be remembered that the CL vs H)~ per-o 0

formance of the Al-~ydrofoil tends to a critical line of CL const. in a

limiting case of H ~ 0, and also that the A2-hydrofoil has a stable per­

formance. Hence, a theoretical possibility of existence of hydrodynamically ,~

stable CL vs H performance is found in the combined hydrofoil.

The above discussion covers only two elementary hydrofoils, Al and

A2 , and only one combination ratio, 1:1. By using more elementary hydro­

foils, A)' A4 , .•• , and more combination ratios, an infinite number of

hydrodynamically stable hydrofoils can be obtained by the method discussed

above. l Furthermore, as indicated in Fig. 5 (which shows the change in the

spray angle T due to changes in dT

n An' the steeper the slope dH ':' at

* H ), the larger the suffix number n of

* H = O. Therefore, a larger n can be

expected to improve the CL vS H performance in the elementary hydrofoils

A.y A4 ,

lEven if the cavity thickness locally becomes negative, the thickness can be increased with very small effect on the CL vS H* performance by means

of slightly increasing (see Fig. 15).

A , o

under an assumption of very small submergence H

24

VII. CONCLUSIONS

The results obtained in this paper may be summarized as follows:

1. An accurate method of estimating the performance of two-dimensional super cavitating hydrofoils of quite arbitrary form operating at quite arbitrary submergence (direct method) through improvement of Johnson's approximate method [5J was proposed. This analysis was based on the following two phys­ically necessary and sufficient conditions for insuring supercavitating flow around the foils:

2.

3·

4.

(i) The absolut~ value of the surface velocity on the pressure side not only needs to be less than that of the cavity surface velocity to avoid oc­currence of cavitation on this side, but also needs to be more than zero.

(ii) The distance between the vacuum-side free streamline and the pressure-side foil surface line, that is, the so-called cavity thickness, should be positive at any point along the chord.

The analysis proposed above may also be applied to obtain hydrofoil shapes for given surface ve­locity distributions (inverse method) without any modification.

The existence of hydrodynamically stable hydro­fo'ils in which the lift coefficient increases as the submergence becomes greater, in the shallower submergence range, was found theoretically pos­sible.

An infinite number of zero form-drag supercavitat­ing hydrofoils of finite lift coefficient CL

(CL < CL < 1, where CL has a specific max max'

value for each hydrofoil) should exist.

The relation between lift coefficient and gence varies considerably among hydrofoil For example, in the flat plate hydrofoil,

* *

submer­shapes. CL at

H = 0 is double its value at H = (l), while, in the Al-hydrofoil of an approximate circular

arc camber, CL changes very little in the same

* working range of 0 < H < (l) (see Figs. 8, 12, and 14).

5. The change in CL . due to change in submergence

corresponds to the change in the surface velocity on the front half of the foil. Therefore, it is quite natural that the change in CL is excep-

tionally large for the flat plate, in which the

4-

surface velocity distribution is one-sided toward the front half of the foil, and fairly small for the Al-hydrofoil, in which the surface velocity

distribution is almo st flat (see Fig. 12). There­fore, it may be expected that the more distorted the hydrofoil camber is toward the rear half of the foil, the less CL will change.

6. Depending on shape of hydrofoil, anyone of three different relationships may exist between drag coefficient and submergence: one in which CD

>[<

decreases with an increase in H as in the flat plate, the second in which CD is con~tant except in the immediate neighborhood of H :::: 0 as in the SHl hydrofoil, and the third in which CD in-

>[<

creases with H as in the SH2 hydrofoil (see Figs. 8 and 9).

7. The drag-lift ratio, or rather· the relationship between the losses and the effective forces of a hydrofoil, changes considerably with submergence and also with shape as shown in Figs. 8 and 9. For example, in the Al-hydrofoil for fixed lift

>[<

coefficient, the drag coefficient at H = co re-duces to about 1/2, 1/5, and l/co for H'~ = 1, 0.25, and 0 respectively.

8. Changes due to submergence are especially impor-_ tant at the shallower submergences of H* < 1 and are not very important in the practical usable range of H* = 1 to 10. However, the difference between the forces in the range of H* = 1 to 10 and the ones at H* = co, is not necessarily neg­ligibly small as shown in Figs. 8 and 9.

9. To check the accuracy of the present solution, a comparison was made with Green's exact solution [lJ for the flat plate operating at an arbitrary submergence as well as the author's second order solution [lOJ for arbitrary form hydrofoils in infinite fluid. The solution was found suffi­ciently accurate. It was also found that the pres­ent solution includes as special limiting cases Tulin-Burkhart's solution in infinite fluid, Wag­ner's solution [12J for planing (H* = 0), and Auslaender's [) ] and Hsu' s [13 J solutions for flat plate hydrofoils operating at arbitrary submer­gences.

25

26

ACKNOWLEffiMENTS

This research has been supported by the Office of Naval Research of

the United States Department of the Navy under Contract Nonr 710(24-), Task

NR 062-052.

The author would like to express his cordial appreciation to Profes­

sors E. Silberman and C. S. Song, St. Anthony Falls Hydraulic Laboratory, Uni­

versity of Minnesota, for useful discussions and encouragement, and also to

Mrs. Mary Marsh and. Mr. Alwin C. H. Young for their help. The manuscript was . prepared for printing by Marjorie Olson.

27

LIST OF REFERENCES

[lJ Green, A. E., "Note on the Gliding of a Plate on the Surface of a Stream," Proceedings Cambridge Phil. Society, Vol. 32, Part 2, 1936, p. 248.

[2J Auslaender, J., "Super cavitating Foils with Flaps Beneath A Free Sur­face," Journal of Basic Engineering, Transactions ASME, Ser­ies D, Vol. 86, 1964, p. 197.

DJ Auslaender, J., "The Lineariz.ed Theory for Super cavitating Hydrofoils Operating at High Speeds Near a Free Surface," Journal of Ship Research, Vol. 6, No.2, 1962, p. 8.

[4 J Luu, T. S. and Fruman, D., "Hydrodynamics, Method for the Design of Superoavitating Hydrofoil Sections in the Presence of Free Surfaoe," Bureau of Ships (Washington, D. C.) Translation No. 885, November 1964.

[5J Johnson, V. E. Jr., Theoretical and Experimental Investigation of Super­cavitating Hydrofoils Operating Near the Free Water Surface, NASA TR R-93, 1961.

[6J Riegels, F., "Das Umstr'omungsproblem bei inkompressiblen Potential­str'omungen," Ing. Arch., Bd. 16, 1948, s. 373; Bd. 17, 1949, S. 94.

[7J Auslaender, J., Low Drag Super cavitating Hydrofoil Sections, Hydronaut­ics Inc., TR 001-7, 1962.

[8J Oba, R., "Theory on Super cavitating Hydrofoils at Arbitrary Cavitation Coefficient," Re orts Institute of Hi h Seed Mechanics Ja­pan, Vol. 15, 1963 1964, p. 1.

[9 J Oba, R., "Lineariz.ed Theory of Supercavi tating Flow Through an Arbi­trary Form Hydrofoil," Zeitschrift fur Angewandte Mathematik und Mechanik, Bd. 41, 1961, s. 354.

[lOJ Oba, R., "Theory for Super cavitating Flow Through an Arbitrary Form Hydrofoil," Journal of Basio Engineering, Transaotions ASME, Series D, Vol. 86, 1964, p. 285.

[llJ Tulin, M. P. and Burkhart, M. P., Linearized Theory for Flows about Lift­ing Foils at Zero Cavitation Number, David Taylor Model Basin Report C-638, 1955.

[12J Wagner, H., Planing of Wateroraft, NACA TM No. 1139, 1948 (Translation "Jahrbuoh der Sohiffbauteohnik," Bd. 34, 1933).

[13J Hsu, C. C., Non-steady Hydrodynamic Characteristios of a Superoavitating Hydrofoil under a Free Surfaoe, Hydronautios Ino. TR 463-2, 1964.

[14 J Weioker, D. , "Dber Sohraubenpropeller fur sehr Sohnelle Schiffe," Schiff und Hafen, Bd. 7, 1959, S. 599.

I "

!

I "

I!£!1LE§2 (1 through 16)

I

1

.. 1

H

y

y Free Surface

Foil

Cavity

0

Fig. 1 - Physical Plane, z = x + iy

~s =0 ·0

~s=O 0

x.

B

~s =0 C--------~~--+-------------------~-----------------O

~------------ ~s=O .. 1

Fig. 2 - Mapping Plane by Riegels' Transformation, z =x + i'y

-ex:> -0 0

C 0 0

.pS(Or4>sl~ 4>.=0 ~s=O .. Fig. 3 - Mapping Plane (Lower Half Plane) I r = ~ + i1]

31

ex:>

B e •

32

tan'l'n

An

2

o o 0.2 0.4 0.6 0.8 1.0

Xs

Fig. 4 - Accuracy of Simple Approximation for yffor Flat Platei

Ao"f 0, Al = A2 = ... = 0

5 r--r-----.--------~----------------------~

Free

~y sur. faces .

_ ...... 1__ Cavity H T X

-f-°h-I·~

4 r---~~-r------~

2 r-----~r-----_4----~=--------r------~

* Fig. 5 - Change in Spray Angle T due to Submergence H for n

Vari ous Shape Parameters A .n

i '

_!S A,

1.0 r--"'--'-~---'-~~---r:::=----_;;::r-----,-------,.

W 0.5~~~--4-----~-~~~~~~~~~~~~~ Y -A,

0.2 0.4 0.6 0.8 1.0

*,..., Fig. 6 - Zero Form-Drag Supercavitating Hydrofoil No.1 at H = 0;

CL ~ i Al , CD';;' 0, (Al 10, Anll = 0)

1.0 r---~""",-----;--""",,----..------r----~

"'s - •• 5AI

.r y

1.5 A, O·~-----~-------+--------r----~~~~~~~

-0.50 0.2 0.4 0.6 0.8 1.0

*,...... Fig. 7 - Zero Form-Drag Supercavitating Hydrofoil No.2 6t H = 0;

CL~~AF CD';;' 0, (A110, A210, Anll ,2=0)

33

34

Co Tl'A2 21

0.2

I [CL ) 10 tAlo.i

o o

I I

/ / --- V

------ Co

~ 1rA2 2"1

/ V

I [CL ] 10{AI

2 4 6 8 10

0.\0 r----~_r_-__r_-__r_---___r_:.,...._,--___,

0.08~-~-+-+---4---L~~--~y

I ./

./

o .04 ~--f--+-+--TT--_'+------;:-"I--t+----l

o o

0.2 0.6 0.8

H~o:,J

H~CXl J

Fig. 8 - Performance of A1-Hydrofoil at Deeper Submergence

12

I ~

1.0

~\ ... \ ........ SHI \ -- r---- ~-" CL 0.8

-=--r ~\ ' ..

CI- H itO

0.6

0.2 o

\ -

1\ --- 1---- _ .. SH2 --,...., ..... 1-- ---

\ \. ~ Flat ' ....... Cu-l'.o

Plate \ ......... f-_ -~----- ---~. t-----

~ SHI

SH2

I(' \ ~~ 2CDH~<X>

2 4 6 8

0.10 r------r-------y--.------r----.-----~_____,

I

I I H*=OO

I

0.08 I----I---.--+----J---,j~---+--+-____,_F_-

0.04 I---I--I---I-L--L-----#-

- .... _ ........ H';:~-

1-",;, - - -H =00 --

12

o 0.2 0.6 0.8

Fig. 9 - Performance of SHl and SH2 Hydrofoils as a Function of * Submergence H

35

36

Yf

CLH*=O

0.4

* H=O SHl

0.2 r---------+---------~----~~~----~ __ =r--------_;

o

0.6

0.4

0.2

0

-0.2

Foil Surface Li ne

o 0.2 0.4 0.6 0.8

SH2 * H =0.5

H*= 0

-.

Foil Surface .......

o 0.2 0.4 0.6 0.8

Fig. 10 - Hydrofoil Shapes and Suction-Side Free Streaml ine Shapes *

1.0

1.0

Yf for Various Submerg.ences H of SHl and SH2 Hydrofoils

0,8 .~----~-------r------.------.-------,

0,6

0.2

o o 0.2 0.4 0.6 0.8 1.0

1.0 r-r------,--------.---------.---------,--------,

0.8 ~~~-+------~------+-----~------~

0.6

0.4

0.2

o o 0.2 0.4 0.6 0.8 1.0

Xs

Fig. 11 - Surface Velocity Distribution Us for Various * Submergences H of SH 1 and SH2 Hydrofoi Is

37

38

-us AIH*:O

(a)

1.0

~ -* '* AI (H:: 0.25) \

\

'" AI

" ..........

~~~-- Flat r-.- .... _ Plate - --~ ... -.--

0.8

0.6 o 0 .. 2 0.4 0.6 0.8 1.0

(b)

2.0

1.0

0 0 0.2 0.4 0.6 0.8 1.0

(c)

0.8

..r=o

0.4 ~-------r------~~~~--;-=-~--~--~--~

o

-0.4 ~ ______ ~ ______ ~~ ______ ~ ______ ~ ______ --J

o 0.2 0.4 0.6 0.8

Fig. 12 - Performance of the Al Approximate Circular Arc

Hydrofoil

1.0

6 ~-------,---------.--------~--------~------~

5 ~r------r--------+-------~~------~--------~

4 ~~-----+--------4---------~-------+~----~4

3 ~~~--~---------+--------~--------+-------~

2

o o 0.2 0,4 0.6 0.8

Fig. 13 - Surface Velocity Distribution u of the Flat Plate s Hydrofoil

1.0

39

40

0.2 ~--------r---------~--------r-------~r-------~

o .~ a 0.2 0.4 0.6 0.8 1.0

1.0

Us . o A2if=o

-1.0 0 0.2 0.4 0.6 0.8 1.0

Xs

0.4 ..... -.!.L

#>~ A2H~O

Yf 0

A2..tO H :: 1.0 0.25

-0.4 o 0.2 0.4 0.6 0.8 1.0

Fig. 14 - Performances of the A21 S-Cambered Hydrofoil

"

0.4 r-------,-------.---------,r-------.------~

0.3 1-___ --1-___ -I-_----;;;~~L......-----L----_l

Free Surface ,-0.2 1----+~-----4

Flot Plate

0.1

o 0.2 0.4 0.6 0.8 H'«- 1.0

Fig. 15 ... Relationship between Cavity Thickness T and the Change in * Lift Coefficient as Related to Submergences H for the

Flat Plate

1.6 .------,---------,,-------.---------,,--------,

1.2 I-___ -I-___ -+-__ -=--~===_~=..L---__I

H=I.O ro III ~ Authors sol. \....,

~ ~~ ~ 0.8 I------+----I------+-----:::;--""~--=,......,=--I ->­o ... a.

(f) 0.5

0.4 I------+----~------~--------,----~

o

o Incidence ongle Ao

Fig. 16 - Comparison between the Calculated Spray Thickness 0 and the Experimental One by Johnson

41

-------------------------------------_._-------.--_.-

APPENDIX

45

APPENDIX

An attempt will be made to compare the solution presented in this paper

with Green's exact solution [lJ for flat plate hydrofoils operating at arbi­

trary submergences.

If the incidence angle A is small of the first order OCE), a

In the physical sense, Green's parameter b corresponding to the sub­

mergence H and/or the spray thickness 6, may be expressed by the parameter

a as follows:

b = 1 + 2a

. . b - ~b2'_ 1 = 1 + 2a - 2a ,/1 + ~ = al

Green's force parameter K is

K = (b - ,1b2 - 1) sin Ao + ~ [2 cos Ao + (b cos Ao - 1) log ~~iJ

1 - cos A 1 __ -::--_-=-0 + 0 ( " 2) - H* 6 = - (b - cos A ) = H + ~ K a K

Notice that 1 - cos A -----K-~o '" O( E ) , as K --- oC E ) • The

(a)

H vs 6 relation was es-

tima ted and is shown in Fig. 16 in parametric form in which Johnson's experi­

mental values [5J are also referred to. Comparing the calculated values and

the experimental ones, the accuracy of Eqs. (a) and (b) seems to be suffi-a ciently good for practical incidence angles of A < 20. If

a 1 -

angle is quite small C for example, A < 50), the value of o

the incidence cos A

a maybe K

46

fairly small. Thus the assumption H;?' H* (see Refs. [2J[3J[5J) might be sub­

stantiated.

Since the disturbance in the free water level associated with the pres­

ence of the hydrofoil (and the spray angle) is related to the shape parameter

* A (see Fig. 5), the H vs H relation for arbitrary shape hydrofoils might o

be estimated approximately by Eqs. (a) and (b) by picking up only the Ao

term. >:.:

Assuming that the modified submergence H of this paper is equal to

the spray thickness (), the lift and the drag coefficients CL and CD are

C L exact 2(b -~ . A A = K Sln 0 cos 0 =

C = 1 C + O(E 3) D exact J D linearized

where

Therefore, if these linearized solutions are modified by a correction factor

J as mentioned above, they agree with Green's exact solutions, up to second

order terms.

I

r

I ..

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Mr. P. Eisenberg, President, Hydronautics, Incorporated, Pindell School Road, Howard County, Laurel, Maryland.

Professor Carl Prohaska, Hydro-og Aerodynamisk Laboratorium, Lyngby, Denmark.

Professor L. Landweber, Iowa Institute of Hydraulic Research, State University of Iowa, Iowa City, Iowa.

Professor H. Rouse, Iowa Institute of Hydraulic Research, State University of Iowa, Iowa City, Iowa.

Professor S. Corrsin, Department of Mechanics, The Johns Hopkins University, Baltimore 18, Maryland.

Professor O. M. Phillips, Division of Mechanical Engineering, In­stitute for Cooperative Research, The Johns Hopkins University, Baltimore 18, Maryland.

Mr. Bill East, Lockheed Aircraft Corporation, California Division, Hydrodynamics Research, Burbank, California.

Mr. R. W. Kermeen, Lockheed Missiles and Space Company, Department 81-73/Bldg 538, P. O. Box 504, Sunnyvale, California. Department' of Naval Architecture and Marine Engineering, Room 5-228, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cam­bridge, Massachusetts 02139. Professor M. A. Abkowitz, Massachusetts Institute of Technology, Cambridge 39, Massachusetts. Professor H. Ashley, Massachusetts Institute of Technology, Cam-bridge 39, Massachusetts. '

Professor A. T. Ippen, Massachusetts Institute of Technology, Cam­bridge 39, Massachusetts.

Professor M. Landahl, Massachusetts Institute of Technology, Cam­bridge 39, Massachusetts. Dr. H. Reichardt, Director, Max-Planck Ins'ti tut fur Stromungsfor­schung, Bottingerstrasse 6-8, Gottingen, Germany •

Professor R. B. Couch, UniversityofMichigan, Ann Arbor, Michigan.

Professor W. W. Willmarth, University of Michigan, Ann Arbor, Mich­igan.

-~2- - --- ----- -- -- ---

Copies Organization

1 Midwest Research Institute, 425 Volker Boulevard, KansasCity,Mis­souri, Attn: Library.

1 Director, St. Anthony Falls Hydraulic Laboratory, University of Min­nesota, Minneapolis 14, Minnesota.

1 Dr. C. S. Song, St. Anthony Falls Hydraulic Laboratory, University of Minnesota, Minneapolis 14, Minnesota.

1 Mr. J. M. Wetzel, St. Anthony Falls Hydraulic Laboratory, University of Minnesota, Minneapolis 14, Minnesota.

1 Head, Aerodynamics Division, National Physical Laboratory, Tedding­ton, Middlesex, England.

1 Mr. A. Silverleaf, National Physical Laboratory, Teddington, Middle­sex, England.

1 The Aeronautical Library, National Research Council, Montreal Road, Ottawa 2, Canada.

1 Dr. J. B. Van Manen, Netherlands Ship Model Basin, Wageningen, The Netherlands.

1 Professor John J.' Foody, Chairman, Engineering Department, State University of New York, Maritime College, Bronx, New York 10465.

1 Professor J. Keller, Institute of Mathematical Sciences, New York University, 25 Waverly Place, New York 3, New York.

1 Professor J. J. Stoker, Institute of Mathematical Sciences, New York University, 25 Waverly Place, New York 3, New York.

1 Dr. T. R. Goodman, Oceanics, Incorporated, Technical Industrial Park, Plainview, Long Island, New York.

1 Professor J. William Holl, Department of Aeronautical Engineering, The Pennsylvania State University, Ordnance Research Laboratory, P. O. Box 30, University Park, Pennsylvania.

1 Dr. M. Sevik, Ordnance Research Laboratory, Pennsylvania State Uni­versity, University Park, Pennsylvania.

1 Dr. George F. Wislicenus, Garfield Thomas Water Tunnel, Ordnance Research Laboratory, The Pennsylvania State University, Post Office Box 30, University Park, Pennsylvania 16801.

1 Mr. David Wellinger, Hydrofoil Projects, Radio Corporation of Amer­ica, Burlington, Massachusetts.

1 The RAND Corporation, 1700 Main Street, Santa Monica, California 90406, Attn: Library.

1 Professor R. C. DiPrima, Department of Mathematics , Rensselaer Poly­technic Institute, Troy, New York.

1 Mr. L. M. White, U. S. Rubber Company, Research and Development De­partment, Wayne, New Jersey.

1 Professor J. K. Lunde, Skipsmodelltanken, Trondheim, Norway.

1 Editor, Applied Mechanics Review, Southwest Research Institute, 8500 Culebra Road, San Antonio 6, Texas.

Copies

1

1

1

1

1

1

1

1

1

1

1

1

1

1

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1

.53

Organization

Dr. H. N. Abramson, Southwest Research Institute, 8.500 Culebra Road, San Antonio 6, Texas.

Mr. G. Ran sIeben , Southwest Research Institute, 8.500 Culebra Road, San Antonio 6, Texas.

Professor E. Y. Hsu, Stanford University, Stanford, California.

Dr. Byrne Perry, Department of Civil Engineering, Stanford Univer ... sity, Stanford, California 9430.5.

Dr. J. P. Breslin, Stevens Institute of Technology, Davidson Lab­oratory, Hoboken, New Jersey.

Mr. D. Savitsky, Stevens Institute of Technology, Davidson Labora­tory, Hoboken, New Jersey.

Mr. S. Tsakonas, Stevens Institute of Technology, Davidson Labora­tory, Hoboken, New Jersey.

Dr. JackKotik, Technical Research Group, Inc., Route 110, Melville, New York.

Dr. R. Timman, Department of Applied Mathematics, Technological Uni .. versity, Julianalaan 132, Delft, Holland.

The Transportation Technical Research.Institute, Investigation Of .. fice, Ship Research Institute, 700 Shinkawa, Mitaka, Tokyo-to, Ja­pan.

Dr. Grosse, Versuchsanstalt furWasserbau und Schiffbau, Schleusen­insel im Tiergarten, Berlin, Germany.

Dr. S. Schuster, Director, Versuchsanstalt furWasserbau und Schiff .. bau, Schleuseninsel im Tiergarten, Berlin, Germany.

Technical Library, Webb Institute of Naval Architecture, Glen Cove, Long Island, New York.

Professor E. V. Lewis, Webb Institute of Naval Architecture, Glen Cove, Long Island, New York.

Mr. C. Wigley, Flat 103, 6-9 Charterhouse Square, London E. C. 1, England.

Coordinator of Research, Maritime Administration, 44lG Street NW, Washington 2.5, D. C.

Unclassified Security Classification

DOCUMEMT COMTROL DATA - R&D (Security classification oil/lie, body 01 abstract and Indexlnll annotat/on mUl't be entered when the overall report 18 c/assllied)

1. O~IGINATIN G ACTIvITY (Corporate author) 2 a. REPORT SEC URI TV C I-ASSI FICA TION

St. Anthony Falls Hydraulic Laboratory, Unclassified University of Minnesota 2b. GROUP

3. REPORT TITLE

ON THE EXISTENCE OF ZERO FORM-DRAG AND HYDRODYNAMICALLY STABLE SUPERCAVITATING HYDROFOILS

4. DESCRIPTIVE NOTES (Type 01 report and Inc/uslve date,.)

Final report on this aspect of study 5. AUTHOR(S) (Last name, Ilrst name, Initial)

-Oba, R.

6. REPO RT DATE 7a. 'foTAI- NO. OF PAGES j7b. NO. OF ~;s November 1965 46

ea. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMJ3ER(S)

Nonr 710(24) Technical Paper No. 54-B b. PROJECT NO.

NR 062-052 c. 9b. OTHER REPORT 1'010(5) (Any other numbers that may be aBBI/lned

this report)

d.

10. AVAILABILITY/LIMITATION NOTICES

Qualified requestors may obtain copies of this report from DDC. e

11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY

Office of Naval Research

13. ABSTRACT

The linearized complex acceleration potential is obtained for a hydrofoil

of arbitrary shape in steady motion beneath a free surface with cavity

of infinite length in simple and compact form. Using some numerical results

obtained from the complex potential, it is shown that there exists

theoretically a super cavitating hydrofoil with finite lift coefficient and

zero form drag. It is also shown that there exists theoretically a super-

cavitating hydrofoil with stable characteristics when shallowly SUbmerged;

that is, the lift coefficient increases as the submergence increases.

,

'.

DD FORM , JA N 64 1473 0101-807-6800

Unclassified Security Classification

14.

Uncl@.s sified Security Classification

KEY WORDS

Supercavitating Hydrofoil

Drag Reduction

Stability

LINK A

ROl.E

LINK 8 LINK C

WT ROl.E WT ROl.E WT

INSTRUCTIONS

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Unclassified Security Classification

------

Gn0000~~~0Q~~~~~~0~OO~~$~~~~~'£C •••• P a~oOPcn·r~~~* •• C~~'&~~8~~~~QPOOq~00aa0n~0nnIJ

Technical Paper No. 54, Series B S1. Anthony Falls Hydraulic Laboratory

ON THE EXISTENCE OF ZERO FORM-DRAG AND HY­DRODYNAMICALLY STABLE SUPER CAVITATING HYDRO­FOILS, by R. aba. November 1965. 46 pages incl. 16 illus. Contract Nonr 710(24).

The linearized complex acceleration potential is obtained for a hydrofoil of arbitrary shape in steady motion beneath a free surface with cavity of infinite length in simple and compact form. Using some numerical results obtained from the complex potential, it is shown that there exists theoret­ically a super cavitating hydrofoil with stable cbaracteristics when shallowly submerged; that is, the lift coefficient in­creases as the submergence increases.

Available from S1. Anthony Falls Hydraulic Laboratory, University of Minnesota, at ~1.50 per copy.

Technical Paper No. 54, Series B S1. Anthony Falls Hydraulic Laboratory

ON THE EXISTENCE OF ZERO FORM-DRAG AND HY­DRODYNAMICALLY STABLE SUPERCA VITATING HYDRO­FOILS, by R. aba. November 1965. 46 pages incL 16 illus. Contract Nonr 710(24).

The linearized complex acceleration potential is obtained for a hydrofoil of arbitrary shape in steady motion beneath a free surface with cavity of infinite length in simple and compact form. Using some numerical results obtained from the complex potential, it is shown that there exists theoret­ically a super cavitating hydrofoil with stable characteristics when shallowly submerged; that is, the lift coefficient in­creases as the submergence increases.

Available from S1. Anthony Falls Hydraulic Laboratory, University of Minnesota, at ~1.50 per copy.

1. Supercavitating Flow 2. Hydrofoil 3. Drag Reduction 4. Zero Drag 5. Stability 6. Acceleration Potential 7. Linear Theory

I. Title II. aba, R.

III. St. Anthony Falls Hydraulic Laboratory

IV. Contract No. Nonr 710(24)

Unclassified

1. Super cavitating Flow 2. Hydrofoil 3. Drag Reduction 4. Zero Drag 5. Stability 6. Acceleration Potential 7. Linear Theory

I. Title II. aba, R.

III. S1. Anthony Falls Hydraulic Laboratory

N. Contract No. Nonr 710(24)

UnclaSSified

Technical Paper No. 54, Series B St. Anthony Falls Hydraulic Laboratory

ON THE EXISTENCE OF ZERO FORM-DRAG AND HY­DRODYNAMICALLY STABLE SUPER CAVITATING HYDRO­FOILS, by R. aba. November 1965. 46 pages incL 16 illus. Contract N onr 710 (24) .

The linearized complex acceleration potential is obtained for a hydrofoil of arbitrary shape in steady motion beneath a free surface with cavity of infinite length in simple and compact form. U sing some numerical results obtained from the complex potential, it is shown that there exists theoret­ically a super cavitating hydrofoil with stable.characteristics when shallowly submerged; that is, the lift coefficient in­creases as the submergence increases.

Available from St. Anthony Falls Hydraulic Laboratory, University of Minnesota, at ~1.50 per copy.

Technical Paper No. 54, Series B st. Anthony Falls Hydraulic Laboratory

ON THE EXISTENCE OF ZERO FORM-DRAG AND HY­DRODYNAMICALLY STABLE SUPER CAVITATING HYDRO­FOILS, by R. Oba. November 1965. 46 pages incL 16 illus. Contract Nonr 710(24).

The linearized complex acceleration potential is obtained for a hydrofoil of arbitrary shape in steady motion beneath a free surface with cavity of infinite length in simple aitd compact form. Using some numerical results obtained from the complex potential, it is shown that there exists theoret­ically a super cavitating hydrofoil with stable characteristics when shallowly submerged; that is, the lift coefficient in­creases as the submergence increases.

Available from St. Anthony Falls Hydraulic Laboratory, UniverSity of Minnesota, at ~1.50 per copy.

1. Supercavitating Flow 2. Hydrofoil 3. Drag Reduction 4. Zero Drag 5. Stability 6. Acceleration Potential 7. Linear Theory

I. Title II. aba, R.

III. St. Anthony Falls Hydraulic Laboratory

N. Contract No. Nonr 710(24)

Unclassified

1. Super cavitating Flow 2. Hydrofoil 3. Drag Reduction 4. Zero Drag 5. Stability 6. Acceleration Potential 7. Linear Theory

1. Title II. aba, R.

III. St. Anthony Falls Hydraulic Laboratory

N. Contract No. Nonr 710(24)

Unclassified

~1

Q0Qg~CO~~Q~c~C~w~~~e~a~~~~~~~~~~~~~~~QQ0000~OOC~QQ~~~~~O •• ~.$ •• ~e~~~Qfl~ge~~~onC00000~ ~ ~n0(0~000 nr0nnor,)O)O

Technical Paper No. 54, Series B St. Anthony Falls Hydraulic Laboratory

ON THE EXISTENCE OF ZERO FORM-DRAG AND HY­DRODYNAMICALLY STABLE SUPER CAVITATING HYDRO­FOILS, by R. aba. November 1965. 46 pages incr. 16 illus. Contract Nonr 710(24).

The linearized complex acceleration potential is obtained for a hydrofoil of arbitrary shape in steady motion beneath a free surface with cavity of infini.te length in simple and compact form. Using some numerical results obtained from the complex potential, it is shown that there exists theoret­ically a super cavitating hydrofoil with stable characteristics when shallowly submerged; that is, the lift coefficient in­creases as the submergence increases.

Available from St. Anthony Falls Hydraulic Laboratory, University of Minnesota, at $1.50 per copy.

Technical Paper No. 54, Series B St. Anthony Falls Hydraulic Laboratory

ON THE EXISTENCE OF ZERO FORM-DRAG AND HY­DRODYNAMICALLY STABLE SUPER CAVITATING HYDRO­FOILS, by R. aba. November 1965. 46 pages incl. 16 illus. Contract Nonr 710 (24).

The linearized complex acceleration potential is obtained for a hydrofoil of arbitrary shape in steady motion beneath a free surface with cavity of infinite length in simple and compact form. Using some numerical results obtained from the complex potential, it is shown that there exists theoret­ically a super cavitating hydrofoil with stable characteristics when shallowly submerged; that is, the lift coeffiCient in­creases as the submergence increases.

Available from St. Anthony Falls Hydraulic Laboratory, University of Minnesota, at $1.50 per copy.

1. Supercavitating Flow 2. Hydrofoil 3. Drag Reduction 4. Zero Drag 5. Stability 6. Acceleratiorr Potential 7. Linear Theory

I. Title II. aba, R.

III. St. Anthony Falls Hydraulic Laboratory

IV. Contract No. Nonr 710(24)

Unclassified

1. Super cavitating Flow 2. Hydrofoil 3. Drag Reduction 4. Zero Drag 5. Stability 6. Acceleration Potential 7. Linear Theory

I. Title II. aba, R.

III. St. Anthony Falls Hydraulic Laboratory

IV. Contract No. Nonr 710(24)

Unclassified

Technical Paper No. 54, Series B St. Anthony Falls Hydraulic Laboratory

ON THE EXISTENCE OF ZERO FORM-DRAG AND HY­DRODYNAMICALL Y STABLE SUPERCA VITATING HYDRO­FOILS, by R. aba. November 1965. 46 pages incl. 16 illus. Contract Nonr 710(24).

The linearized complex acceleration potential is obtained for a hydrofoil of arbitrary shape in steady motion beneath a free surface with cavity of infinite length in simple and compact form. Using some numerical results obtained from the complex potential, it is shown that there exists theoret­ically a super cavitating hydrofoil with stable.characteristics when shallowly submerged; that is, the lift coefficient in­creases as the submergence increases.

Available from St. Anthony Falls Hydraulic Laboratory, University of Minnesota, at $1.50 per 'copy.

Technical Paper No. 54, Series B St. Anthony Falls Hydraulic Laboratory

ON THE EXISTENCE OF ZERO FORM-DRAG AND HY­DRODYNAMICALLY STABLE SUPER CAVITATING HYDRO­FOILS, by R. Oba. November 1965. 46 pages incl. 16 illus. Contract Nonr 710(24).

The linearized complex acceleration potential is obtained for a hydrofoil of arbitrary shape in steady motion beneath a free surface with cavity of infinite length in simple aJ.i.d compact form. Using some numerical results obtained from the complex potential, it is shown that there exists theoret­ically a super cavitating hydrofoil with stable characteristics when shallowly submerged; that is, the lift coefficient in­creases as the submergence increases.

Available from St. Anthony Falls Hydraulic Laboratory, University of Minnesota, at $1.50 per copy.

-~ ~~0~nrOQ~Qe~.~~~~.

1. Supercavitating Flow 2. Hydrofoil 3. Drag Reduction 4. Zero Drag 5. Stability 6. Acceleration Potential 7. Linear Theory

I. Title II. aba, R.

III. St. Anthony Falls Hydraulic Laboratory

IV. Contract No. Nonr 710(24)

Unclassified

1. Supercavitating Flow 2. Hydrofoil 3. Drag Reduction 4. Zero Drag 5. Stability 6. Acceleration Potential 7. Linear Theory

I. Title II. aba, R.

III. St. Anthony Falls Hydraulic Laboratory

IV. Contract No. Nonr 710(24)

Unclassified

')

n~n~D~~~~~~~.~.$&~P~~.~g.~ •• ~~~r~·~.~~.~.~~~~~~~~.~~~~~O.~.~ •• * •• e~~~~~~~~eC~QQ~O~

Technical Paper No. 54, Series B St. Anthony Falls Hydraulic Laboratory

ON THE EXISTENCE OF ZERO FORM-DRAG AND HY­DRODYNAMICALLY STABLE SUPER CAVITATING HYDRO­FOILS, by R. aba. November 1965. 46 pages incr. 16 illus. Contract Nonr 710 (24).

The linearized complex acceleration potential is obtained for a hydrofoil of arbitrary shape in steady motion beneath a free surface with cavity of infinite length in simple and compact form. Using some numerical results obtained from the complex potential, it is shown that there exists theoret­ically a supercavitating hydrofoil with stable characteristics when shallowly submerged; that is, the lift coefficient in­creases as the submergence increases.

Available from St. Anthony Falls Hydraulic Laboratory, University of Minnesota, at $1.50 per copy.

Technical Paper No. 54, Series B St. Anthony Falls Hydraulic Laboratory

ON THE EXISTENCE OF ZERO FORM-DRAG AND HY­DRODYNAMICALLY STABLE SUPERCAVITATING HYDRO­FOILS, by R. aba. November 1965. 46 pages incl. 16 illus. Contract Nonr 710 (24).

The linearized complex acceleration potential is obtained for a hydrofoil of arbitrary shape in steady motion beneath a free surface with cavity of infinite length in simple and compact form. Using some numerical results obtained from the complex potential, it is shown that there exists theoret­ically a super cavitating hydrofoil with stable characteristics when shallowly submerged; that is, the lift coefficient in­creases as the submergence increases.

Available from St. Anthony Falls Hydraulic Laboratory, University of Minnesota, at $1.50 per copy.

1. Supercavitating Flow 2. Hydrofoil 3. Drag Reduction 4. Zero Drag 5. Stability 6. Acceleration Potential 7. Linear Theory

I. Title II. Oba, R.

III. St. Anthony Falls Hydraulic Laboratory

IV. Contract No. Nonr 710(24)

Unclassified

1. Super cavitating Flow 2. Hydrofoil 3. Drag Reduction 4. Zero Drag 5. Stability 6. Acceleration Potential 7. Linear Theory

I. Title II. Oba, R.

III. St. Anthony Falls Hydraulic Laboratory

IV. Contract No. Nonr 710(24)

Unclassified

Technical Paper No. 54, Series B St. Anthony Falls Hydraulic Laboratory

ON THE EXISTENCE OF ZERO FORM-DRAG AND HY­DRODYNAMICALL Y STABLE SUPER CAVITATING HYDRO­FOILS, by R. Oba. November 1965. 46 pages inc!. 16 illus. Contract Nonr 710 (24).

The linearized complex acceleration potential is obtained for a hydrofoil of arbitrary shape in steady motion beneath a free surface with cavity of infinite length in simple and compact form. Using some numerical results obtained from the complex potential, it is shown that there exists theoret­ically a super cavitating hydrofoil with stable.characteristics when shallowly submerged.; that is, the lift coefficient in­creases as the submergence increases.

Available from St. Anthony Falls Hydraulic Laboratory, University of Minnesota, at $1.50 per copy.

Technical Paper No. 54, Series B St. Anthony Falls Hydraulic Laboratory

ON THE EXISTENCE OF ZERO FORM-DRAG AND HY­DRODYNAMICALL Y STABLE SUPERCA VITATING HYDRO­FOILS, by R. Dba. November 1965. 46 pages inc!. 16 illus. Contract Nonr 710(24).

The linearized complex acceleration potential is obtained for a hydrofoil of arbitrary shape in steady motion beneath a free surface with cavity of infinite length in simple aitd compact form. Using some numerical results obtained from the complex potential, it is shown that there exists theoret­ically a super cavitating hydrofoil with stable characteristics when shallowly submerged; that is, the lift coefficient in­creases as the submergence increases.

Available from St. Anthony Falls Hydraulic Laboratory, University of Minnesota, at $1.50 per copy.

1. Supercavitating Flow 2. Hydrofoil 3. Drag Reduction 4. Zero Drag 5. Stability 6. Acceleration Potential 7. Linear Theory

1. Title II. Oba, R.

III. St. Anthony Falls Hydraulic Laboratory

IV. Contract No. Nonr 710(24)

Unclassified

1. Supercavitating Flow 2. Hydrofoil 3. Drag Reduction 4. Zero Drag 5. Stability 6. Acceleration Potential 7. Linear Theory

I. Title II. aba, R.

III. St. Anthony Falls Hydraulic Laboratory

IV. Contract No. Nonr 710(24)

Unclassified